The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

64 3. The derivation of nuclear forces from chiral Lagrangians
form we should require the same transformation property (3.104) under go E SU(2)v X SU(2)A'
Using eqs. (3.98), (3.100) and (3.101) one verifies that uJ.L indeed satisfies this condition, i.e. that
and thus corresponds to the matrix representation of the covariant derivative of �.
(3.105)
The pion fields can be defined by a particular parametrization of U. Choosing again the 2 x 2
matrix representation (3.78) of the group generators Ai, Vi we can parametrize U, for example,
as follows:
Z'Fr • T
U=exP ( h ) , (3.106)
where 7ri are pseudoscalar fields (pions) and f1f is some constantP This also uniquely defines the
quantity uJ.L in terms of the pion fields. It is easy to construct chiral invariant terms consisting of
u/s. One simply has to build traces ( ... ) in flavor space of any product of the uJ.L's. For instance,
the term (uJ.Lu J.L ) is clearly chiral invariant:
(3.107)
Let us now introduce the nucleon field \]J. We will require that the nucleon field belongs to the
CCWZ realization and transforms under go E G via eq. (3.54):
(3.108)
Since we have chosen the representation (3.78) of the group generators, h(go, 'Fr) is a 2 x 2 matrix
in the flavor space. Note that h(go, 'Fr) becomes independent on 'Fr for all go E SU(2)v and forms
the fundamental representation of SU(2)v. To define the covariant derivatives of the nucleon field
it is convenient to introduce the so-called connection r J.L:
It follows from eq. (3.98) that r J.L transforms as:
Here we made use of the similar trick as in eq. (3.103):
Therefore, the covariant derivative of the nucleon defined as
transforms in the same way as the field \]J (covariantly):
(3.109)
(3.110)
(3.111)
(3.112)
(3.113)
Thus, any flavor scalar built up from isospinors \]J, D J.L \]J with insertions of uJ.L is chiral invariant.
17 It can be shown that in the chiral limit f" is the pion decay constant.